Writer: Ken Mirasol
Fibonacci Sequence
Learning Objectives:
At the end of the lesson, you should be able to:
§
define Fibonacci sequence;
§
determine the relationship between the Fibonacci sequence and the number Φ (Phi);
§
find the nth term of a Fibonacci sequence;
§
determine the relationship between Fibonacci sequence and Pythagorean theorem; and
§
generate Pythagorean triples using four consecutive terms of Fibonacci sequence.
Getting Connected
In 1202, an interesting sequence of numbers aroused from one of the problems
posed in a book Liber Abaci by Italian mathematician Leonardo Pisano Bigollo
(1170-1250) also known as Leonardo of Pisa. The problem goes like this:
The Rabbit Problem
Beginning with a single pair of rabbits, a male and a female, how many
pairs of rabbits will be born in a year, assuming that every month, each
pair of male and female rabbits breeds a new pair of male and female
rabbits and the new pair of rabbits also starts breeding additional pairs
of rabbits after the first month of their birth?”
To answer the question, let us illustrate the problem below.
January: 1
February: 1
March: 2
April: 3
May: 5
First Quarter: Fibonacci Sequence
Looking at the number of pair of rabbits, what did you observe about the
relationship of the number of pairs on March as compared to the number of
pairs on January and February? How about from the month of April to the
month of March and February? Were you able to see a pattern? The number
of pair of rabbit is equal to the sum of the number of pair of rabbits on the two
previous months. Listing all the pairs per month, we have:
Jan
1
Feb
1
Mar
2
Apr
3
May
5
Jun
8
Jul
13
Aug
21
Sep
34
Oct
55
Nov
89
Dec
144
Jan
233
Hence, there will be 233 pairs of rabbits after a year. The above sequence of
numbers is called a Fibonacci sequence.
Fibonacci Sequence
A sequence of numbers in which the first two terms are both 1 and each
term thereafter is obtained by adding the preceding two terms.
Learning Tip: Leonardo Pisano Bigollo is popularly known as Fibonacci which also means
“son of Bonacci.”
The Fibonacci Sequence and the Number 𝚽 (Phi)
It is interesting to note that when we get the ratio of the two consecutive terms of the
Fibonacci sequence it is approaching the number Φ (Phi), also known as the Golden ratio,
which is approximately equal to 1.618034.
𝐹! 1
=
𝐹" 1
=
1
𝐹# 13
=
𝐹$
8
=
1.6250
𝐹% 2
=
𝐹! 1
=
2
𝐹& 21
=
𝐹# 13
=
1.6154
𝐹' 3
=
𝐹% 2
=
1.5
𝐹( 34
=
𝐹& 21
=
1.6190
𝐹) 5
=
𝐹' 3
=
1.6667
𝐹"* 55
=
𝐹(
34
=
1.6177
𝐹$ 8
=
𝐹) 5
=
1.6
𝐹"" 89
=
𝐹"* 55
=
1.6182
First Quarter: Fibonacci Sequence
The nth Term of Fibonacci Sequence
Unlike the arithmetic sequence and geometric sequence where we can derive
the nth term, deriving the nth term of the Fibonacci sequence will be tedious.
However, we can find a specific term of the Fibonacci sequence using Φ.
The nth Term of Fibonacci Sequence
The nth term Fn of the Fibonacci sequence using the golden ratio number
Φ, which is equal to 1.618034, is given by
Φ! − (1 − Φ)!
𝐹! =
√5
Example 1: Find the 20th term of the Fibonacci sequence.
Solution: We substitute 𝑛 = 20 and Φ = 1.618034 to get the 20th term of the
Fibonacci sequence.
𝐹! = Φ! − (1 − Φ)!
√5
𝐹"# = (1.618034)"# − (1 − 1.618034)"#
√5
𝐹"# = 610
Example 2: Find the 15th term of the Fibonacci sequence.
Solution: We substitute 𝑛 = 15 and Φ = 1.618034 to get the 15th term of the
Fibonacci sequence.
𝐹! = Φ! − (1 − Φ)!
√5
𝐹$% = (1.618034)$% − (1 − 1.618034)$%
√5
𝐹$% = 6765
Your Turn: Find the following terms of Fibonacci sequence using the number Φ.
1. 𝐹!
2. 𝐹"#
3. 𝐹$%
First Quarter: Fibonacci Sequence
The Fibonacci Sequence and Pythagorean Theorem
One of the interesting discovery in the field of Geometry is the Pythagorean
Theorem. It states that given a right triangle with legs a and b, and hypotenuse
c, the sum of the squares of the lengths of two legs is equal to the square of
the hypotenuse. In symbols,
a2 + b2 = c2
Using the four consecutive terms of the Fibonacci sequence, we can generate
triples that will satisfy the Pythagorean Theorem. Follow the steps below.
Generating Pythagorean Triples Using Terms of Fibonacci Sequence
Choose any four consecutive terms of the Fibonacci sequence. Then,
1. Let a be the product of the first and last terms of the four numbers.
2. Let b be twice the product of the two middle terms.
3. Let c be equal to the square root of the sum of the square of a and b.
Example 1: Generate a Pythagorean triple using the following terms of the
Fibonacci sequence.
1, 1, 2, 3
Solution:
Solving for a:
𝑎 =1⋅3=3
Solving for b:
𝑏 = 2(1 ⋅ 2) = 2(2) = 4
Solving for c:
𝑐 = 8(3)$ + (4)$ = √9 + 16 = √25 = 5
Hence, the Pythagorean triple generated from the given four consecutive
Fibonacci numbers are {3, 4, 5}.
Learning Tip: If (a, b, c) are Pythagorean triples and the greatest common divisor of a, b,
and c is 1, then (a, b, c) is said to be a primitive triple.
First Quarter: Fibonacci Sequence
Example 2: Generate a Pythagorean triple using the following terms of the
Fibonacci sequence.
2, 3, 5, 8
Solution:
Solving for a:
𝑎 = 2 ⋅ 8 = 16
Solving for b:
𝑏 = 2(3 ⋅ 5) = 2(15) = 30
Solving for c:
𝑐 = 8(16)$ + (30)$ = √256 + 900 = √1156 = 34
Hence, the Pythagorean triple generated from the given four consecutive
Fibonacci numbers are {16, 30, 34}.
Your Turn: Generate a Pythagorean triple using the following four terms of Fibonacci
sequence.
1. 1, 2, 3, 5
2. 3, 5, 8, 13
3. 8, 13, 21, 34
First Quarter: Fibonacci Sequence